IB DP Math AI- Topic : AHL 1.11 -infinite geometric sequences HL Paper 2

Question

The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8.

a.Find the first term and the common difference.[4]

 

b.Find the smallest value of n such that the sum of the first n terms is greater than 600.[3]

 
▶️Answer/Explanation

Markscheme

\({S_n} = \frac{n}{2}[2a + (n – 1)d]\)

\(212 = \frac{{16}}{2}(2a + 15d)\,\,\,\,\,( = 16a + 120d)\)     A1

\({n^{th}}{\text{ term is }}a + (n – 1)d\)

\(8 = a + 4d\)     A1

solving simultaneously:     (M1) 

\(d = 1.5,{\text{ }}a = 2\)     A1

[4 marks]

a.

\(\frac{n}{2}[4 + 1.5(n – 1)] > 600\)     (M1)

\( \Rightarrow 3{n^2} + 5n – 2400 > 0\)     (A1)

\( \Rightarrow n > 27.4…,{\text{ }}(n < – 29.1…)\) 

Note: Do not penalize improper use of inequalities. 

 

\( \Rightarrow n = 28\)     A1

[3 marks]

b.

Examiners report

This proved to be a good start to the paper for most candidates. The vast majority made a meaningful attempt at this question with many gaining the correct answers. Candidates who lost marks usually did so because of mistakes in the working. In part (b) the most efficient way of gaining the answer was to use the calculator once the initial inequality was set up. A small number of candidates spent valuable time unnecessarily manipulating the algebra before moving to the calculator.

a.

This proved to be a good start to the paper for most candidates. The vast majority made a meaningful attempt at this question with many gaining the correct answers. Candidates who lost marks usually did so because of mistakes in the working. In part (b) the most efficient way of gaining the answer was to use the calculator once the initial inequality was set up. A small number of candidates spent valuable time unnecessarily manipulating the algebra before moving to the calculator.

b.

Question

Each time a ball bounces, it reaches 95 % of the height reached on the previous bounce.

Initially, it is dropped from a height of 4 metres.

a.What height does the ball reach after its fourth bounce?[2]

b.How many times does the ball bounce before it no longer reaches a height of 1 metre?[3]

c.What is the total distance travelled by the ball?[3]

▶️Answer/Explanation

Markscheme

height \( = 4 \times {0.95^4}\)     (A1)

= 3.26 (metres)     A1

[2 marks]

a.

\(4 \times {0.95^n} < 1\)     (M1)

\({0.95^n} < 0.25\)

\( \Rightarrow n > \frac{{\ln 0.25}}{{\ln 0.95}}\)     (A1)

\( \Rightarrow n > 27.0\) 

Note: Do not penalize improper use of inequalities.

 

\( \Rightarrow n = 28\)     A1 

Note: If candidates have used n – 1 rather than n throughout penalise in part (a) and treat as follow through in parts (b) and (c).

 

[3 marks]

b.

METHOD 1

recognition of geometric series with sum to infinity, first term of \(4 \times 0.95\) and common ratio 0.95     M1

recognition of the need to double this series and to add 4     M1

total distance travelled is \(2\left( {\frac{{4 \times 0.95}}{{1 – 0.95}}} \right) + 4 = 156{\text{ (metres)}}\)     A1

[3 marks] 

Note: If candidates have used n – 1 rather than n throughout penalise in part (a) and treat as follow through in parts (b) and (c).

 

METHOD 2

recognition of a geometric series with sum to infinity, first term of 4 and common ratio 0.95     M1

recognition of the need to double this series and to subtract 4     M1

total distance travelled is \({\text{2}}\left( {\frac{4}{{1 – 0.95}}} \right) – 4 = 156{\text{ (metres)}}\)     A1

[3 marks]

c.

Examiners report

The majority of candidates were able to start this question and gain some marks, but only better candidates gained full marks. In part (a) the common error was to assume the wrong number of bounces and in part (b) many candidates lost marks due to rounding the inequality in the wrong direction. Part (c) was found difficult with only a limited number recognising the need for the sum to infinity of a geometric sequence and many of those did not recognise how to link the sum to infinity to the total distance travelled.

Question

Find the sum of all the multiples of 3 between 100 and 500.

▶️Answer/Explanation

Markscheme

METHOD 1

102 + 105 + … + 498     (M1)

so number of terms = 133     (A1)

EITHER

\( = \frac{{133}}{2}(2 \times 102 + 132 \times 3)\)     (M1)

= 39900     A1

OR

\( = (102 + 498) \times \frac{{133}}{2}\)     (M1)

= 39900     A1

OR

\(\sum\limits_{n = 34}^{166} {3n} \)     (M1)

= 39900     A1

METHOD 2

\(500 \div 3 = 166.666…{\text{ and }}100 \div 3 = 33.333…\)

\(102 + 105 + … + 498 = \sum\limits_{n = 1}^{166} {3n}  – \sum\limits_{n = 1}^{33} {3n} \)     (M1)

\(\sum\limits_{n = 1}^{166} {3n}  = 41583\)     (A1)

\(\sum\limits_{n = 1}^{33} {3n}  = 1683\)     (A1)

the sum is 39900     A1

[4 marks]

Examiners report

Most candidates got full marks in this question. Some mistakes were detected when trying to find the number of terms of the arithmetic sequence, namely the use of the incorrect value n = 132 ; a few interpreted the question as the sum of multiples between the 100th and 500th terms. Occasional application of geometric series was attempted.

Question

A metal rod 1 metre long is cut into 10 pieces, the lengths of which form a geometric sequence. The length of the longest piece is 8 times the length of the shortest piece. Find, to the nearest millimetre, the length of the shortest piece.

▶️Answer/Explanation

Markscheme

the pieces have lengths \(a,{\text{ }}ar,{\text{ …, }}a{r^9}\)     (M1)

\(8a = a{r^9}{\text{ }}({\text{or }}8 = {r^9})\)     A1

\(r = \sqrt[9]{8} = 1.259922…\)     A1

\(a\frac{{{r^{10}} – 1}}{{r – 1}} = 1\,\,\,\,\,\left( {{\text{or }}a\frac{{{r^{10}} – 1}}{{r – 1}} = 1000} \right)\)     M1

\(a = \frac{{r – 1}}{{{r^{10}} – 1}} = 0.0286…\,\,\,\,\,\left( {{\text{or }}a = \frac{{r – 1}}{{{r^{10}} – 1}} = 28.6…} \right)\)     (A1)

a = 29 mm (accept 0.029 m or any correct answer regardless the units)     A1

[6 marks]

Examiners report

This question was generally well done by most candidates. Some candidates recurred to a diagram to comprehend the nature of the problem but a few thought it was an arithmetic sequence.

A surprising number of candidates missed earning the final A1 mark because they did not read the question instructions fully and missed the accuracy instruction to give the answer correct to the nearest mm.

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